Structure and stability of graphene nanoribbons in oxygen, carbon dioxide, water, and ammonia

We determine, by means of density functional theory, the stability and the structure of graphene nanoribbon (GNR) edges in presence of molecules such as oxygen, water, ammonia, and carbon dioxide. As in the case of hydrogen-terminated nanoribbons, we find that the most stable armchair and zigzag configurations are characterized by a non-metallic/non-magnetic nature, and are compatible with Clar's sextet rules, well known in organic chemistry. In particular, we predict that, at thermodynamic equilibrium, neutral GNRs in oxygen-rich atmosphere should preferentially be along the armchair direction, while water-saturated GNRs should present zigzag edges. Our results promise to be particularly useful to GNRs synthesis, since the most recent and advanced experimental routes are most effective in water and/or ammonia-containing solutions.


I. INTRODUCTION
Narrow graphene nanoribbons 1 are carbon allotropes that promise to combine the high carrier mobility of graphene 2 with a semiconducting nature due to quantum confinement in the lateral direction. The resulting systems are likely to be suitable for direct application as channel in field-effect transistors and other semiconducting devices. The design of GNRs, and more generally of graphene samples with controlled edges, is crucial in determining their electronic and transport properties. Among the variety of reported experimental approaches, three routes, intensely explored in the last couple of years, seem at the frontier of current research: i) lithographic patterning [3][4][5][6] ; ii) sonochemical methods 7,8 ; iii) metallic nanoparticle chemical etching (unzipping) of graphite/graphene (nanotubes) [9][10][11][12][13][14] . Gaining control of the edge profiles and structures is of course one of the most important issues to solve, somehow defining the potential impact of these approaches. The observation of graphene edges 15,16 and/or GNR 7,14 profiles at atomicscale resolution indicates that the mentioned experimental methods are capable of producing smooth, likely nonchiral (i.e. no mixing of zigzag and armchair direction), edges. However, little is known on how to achieve either zigzag or armchair edges, and about the actual chemical termination of the unsaturated carbon bonds.
In this regard, the huge number of theoretical works that recently appeared in the literature on GNRs [17][18][19][20][21][22][23][24][25][26][27] has almost exclusively focused on single-hydrogen saturation of the edge carbon atoms. This was mainly due to the surprising discovery of spin-polarized electronic states localized on the edges of zigzag GNRs 17,18 , possibly turning them into half-metals under high electric fields 19,20 , and spurring great interest in view of possi-ble applications in future spintronics. However, we have recently 28,29 pointed out that the single-H-terminated zigzag GNRs, and thus all those fascinating electronic and magnetic properties they exhibit, might only be stable at very difficult experimental conditions.
The hydrogen case-study is presently extended to GNRs terminated with other relevant atoms and molecules [30][31][32][33][34][35] . In fact, as mentioned above, experiments are often performed in aqueous or ammonia-containing solutions, or in oxidizing conditions. More generally water and oxygen contamination (for example in air) is extremely important in determining the electronic properties of graphitic systems. For example, the chemical reactions that induce the unzipping of carbon nanotubes are thought to leave O-terminated zigzag GNRs as end products 11 . In this work we thus report density functional theory (DFT) calculations of the energetics and structures of various GNR edges terminated with atmospheric molecules, such as oxygen, water, and ammonia, as functions of the respective chemical potentials. The work is organized as follows: in section II we describe the theoretical method and define the relevant thermodynamic quantities; in section III we present our systematic study on oxygen-terminated zigzag and armchair GNRs. In section IV we report our results on selected GNRs in presence of water and ammonia molecules; discussion on the electronic and magnetic properties and comparisons with experimental data are drawn in section V, while section VI is devoted to conclusions and perspectives.

II. METHOD
DFT calculations were performed with the PWSCF code of the Quantum ESPRESSO suite 36 in a plane-arXiv:1009.1808v1 [cond-mat.mes-hall] 9 Sep 2010 wave/ultrasoft-pseudopotential approach, adopting the Generalised Gradient Approximation (GGA) from Ref. 37 for the exchange-correlation functional. We used a kinetic energy/charge cut-off of 30/300 Ry. Edges were simulated within a super-cell geometry using a vacuum layer of 9.5Å between two edges and of 10.6Å between two graphene planes. L was fixed according to the latticeconstant of graphene (2.46Å) and atomic positions were allowed to fully relax. In the case of armchair edges, the width of the ribbons was about 22.4Å and for zigzag ribbons about 28.9Å. Electronic integrations were done using equidistant, consistent grids along the periodic direction of the ribbon corresponding to 12 k-points in the smallest unit cell. Denser grids have been occasionally used in the presence of metallic/magnetic states. We limited our study to the thermodynamic stability of neutral systems, and as such kinetics and charge effects are not included. In fact, although graphene edges and GNR can be obtained experimentally in charged atmospheres, their technological interest resides in their electronic properties in the neutral state, i.e. at a Fermi level as close as possible to the Dirac point of the graphene band structure. We argue that GNRs and graphene edges obtained at high doping level might be extremely reactive when brought back to neutrality, and recombine to form more stable edge structures.

III. OXYGEN AND CARBON DIOXIDE
As in Ref. 28 , we calculate the zero-temperature edge formation-energy per unit length as follows: where E ribb , E blk and E O2 are the total energy of the ribbon super-cell, of one atom in "bulk" graphene and of the isolated O 2 molecule. N C (N O ) are the number of carbon (oxygen) atoms in the super-cell, while L the length of the unit cell (see Fig. 1). E O2 can be used to determine the stability of different structures as a function of the experimental conditions 38 . In presence of molecular O 2 gas, at a given chemical potential µ O2 , the relative stability is obtained by comparing . At the absolute temperature T and for a partial O 2 pressure P 39 , where H • (S • ) is the enthalpy (entropy) at the pressure P • = 1 bar obtained from Ref. 40 . In presence of monoatomic-oxygen gas one should use .86 eV, i.e. the binding energy per atom of the O 2 molecule in DFT-GGA. The most stable structures are then obtained, as in Ref. 28 , by comparing their respective Gibbs free energy G O2 as function of the (molecular) oxygen chemical potential. G is linear in µ, the slope being determined by ρ O . For a given value of µ the most stable structure is the one with the lowest value of G, thus, by increasing µ (i.e. going to an environment richer in oxygen) the favorable structures will be those with higher oxygen-density ρ O . We report in Fig. 1 a pictorial representation of the different GNR edges we have considered in this study, along with their characteristic periodicity L. In the left top line we show the clean zigzag and armchair GNRs 23 . The right top line (second line from top) represent the zigzag (armchair) oxygen-covered nanoribbons included in this work. Structures are chosen as to saturate each oxygen atom with two single bonds or a double one, while other possible chemical bonding configurations are much less stable and are not included in the results. The edges are ordered with respect to increasing oxygen density and, at equal density, with respect to the energetics, from most favorable to least favorable. Non-aromatic edges 28,29 are indicated with a tilde (see section V).
The density and zero-temperature edge-formation energy are reported in Table I. The free-energy G stability diagram is reported in Fig. 2. We note that three zigzag GNRs, namedz o2 , z o4 , and z o5 , and three armchair GNRs, named a o11 , a o12 , and a o13 , have negative free energies, i.e. graphene should spontaneously break and form such ribbons if the experimental conditions allow to overcome the reaction barriers, as for example when nanoparticles, while sliding on graphene layers, are able to cut edges along their path 9,10 . We note that those experiments are performed in hydrogen atmosphere; in the case of oxygen atmosphere the reaction is more exothermic and could therefore be less controllable.
The thermodynamic stability diagram (Fig. 2) indicates that the armchair a o12 and the zigzag z o4 are the most stable configurations at low and high concentrations of atomic or molecular oxygen, respectively. However, the former one is more stable for nearly all negative values of the O 2 chemical potential, which suggests that the armchair edge should be the only thermodynamically stable one in ordinary experimental conditions going from ultra-vacuum to atmospheric concentration of molecular oxygen.
On the other hand, it is even more important to consider the presence of CO 2 molecules, either from the atmosphere or from the oxidation of edge carbon atoms. We can define, analogously to Eq. 1, an edge formation energy per unit length with respect to carbon dioxide, which reads: (3) By simple calculations one can easily show that where ρ O is, as above, the oxygen density per unit length, and ∆E CO2 = E blk + E O2 − E CO2 is the formation energy of molecular CO 2 from graphene and oxygen molecules, and equals -3.76 eV in DFT-GGA. In other words, the chemical reaction C blk + O 2 → CO 2 is strongly exothermic, as well known from the fact that ordinary coal burns in oxygen once ignited. As a consequence, the molecular-oxygen thermodynamic stability scale equally holds for carbon dioxide, with a horizontal shift of ∆E CO2 . Conversely, the molecular oxygen thermodynamic stability scale must include a vertical line at µ O2 = ∆E CO2 = −3.76 eV: at higher oxygen densities no edge decoration can be thermodynamically more stable than CO 2 . This, again, implies that only the armchair a o13 edge is realistically observable at thermodynamic equilibrium and oxygen concentration lower than the one which would induce spontaneous ignition of graphene with oxygen to produce CO 2 .

IV. WATER AND AMMONIA
Besides oxygen, we considered GNR edges saturated with water or ammonia molecules. In these cases we restricted our study to O/H/N contents as fixed by the respective molecular stoichiometries, which corresponds to the description of experiments carried out in pure water or ammonia atmospheres. In principle, the full study of other stoichiometries would allow a more complete description of the thermodynamic properties as function of two independent chemical potentials, i.e. O and H (N and H) in the case of water (ammonia). However, the number of different possible edges is formidable, which would make these calculations extremely demanding, virtually impossible to carry out "by hand", and likely needing random-search optimizations 41 . Moreover, we only considered aromatic edge configurations 28 , with a few exceptions in the case of water. The zero-temperature edge formation-energies per length reads as follows: and in the case of water and ammonia, respectively. The restriction on aromatic edges implies that zigzag edges respect the 2-1-1 bond order rule for the dangling carbon bond that, as shown in Refs. 28,29 , is the only aromatic configuration for zigzag GNRs. In the case of water, single dangling bonds are thus saturated by H atoms or OH groups, while double bonds are saturated either by oxygen atoms, or by H/OH-containing pairs (see Fig. 1, bottom row). In the case of ammonia, we considered all water-decorated structures, and replaced each O atom (OH group), by a NH (NH 2 ) group. The formation energies are summarized in table I, the corresponding thermodynamic stability plots are shown in Fig. 3 and Fig. 4.
In the case of water, two main results are evident: the free energy is always positive, meaning that graphene would not spontaneously break to form edges in a water atmosphere. On the other hand, any freshly-cut edge would be substantially stabilized in the presence of water by the saturation of its dangling bonds with O/H/OH groups. The other result is that z oh1 is the most stable configuration at any chemical potential. This means that, in principle, the presence of oxygen favors the armchair edges, while the presence of water favors the zigzag ones. In the case of ammonia, free energies are always positive, and the z nh1 zigzag GNR is the most stable one, in correspondence to z oh1 in presence of water.

V. DISCUSSION
We have shown in our previous works on hydrogendecorated edges 28 and GNRs 29 that a number of different edge structures with different electronic/magnetic properties compete for thermodynamic stability at different hydrogen concentrations, and that they should be considered in order to provide results of practical use to experimentalists. In particular, we showed that an important hint on the structural and chemical stability of GNR edges comes from the well-known organic chemistry structural properties of polycyclic aromatic hydrocarbons (PAHs), large molecules consisting of fused carbon rings. According to the so-called Clar's rule 42,43 , the most stable structures for PAHs are the ones maximizing the number of Clar sextets.
In bulk graphene all π-electrons take part in the formation of Clar sextets. In this arrangement, one out of 3 hexagons is a Clar sextet. We define, as in Refs. 28,29 , aromatic edges as those allowing for the same 1/3 density of benzenoid rings as bulk graphene, and in which every carbon atom is fully saturated (it has 4 covalent bonds). Conversely, in non-aromatic edges the 1/3 bulkgraphene-density of benzenoid rings is incompatible with a full saturation of all carbon dangling bonds. These edges are indicated with a˜in their label. This density can only be recovered at the price of having carbon atoms with more or less than 4 covalent bonds. In the presence of magnetism such over/under-bonding can be interpreted as radicals, as in the edge labelledã o6 (see Fig. 5, panel B, and discussion in Ref. 29 ).
As shown in Fig. 1 and in Table I, we find that, except forz o2 and analogously to the case of hydrogen-saturated edges, the energetically most stable ones are aromatic. In the case of water, thez oh4 edge is the only relatively stable non-aromatic configuration. In Fig. 5 we report the electronic band structure, close to the Fermi level, of the most stable and interesting edges. The gray-shaded areas indicate the edge-projected bulk bands. The full lines correspond to GNR electronic states. Bands outside the gray-shaded area are electronic states exponentially localized at the edges (edge states).
In line with the results of Ref. 28 , in Table I and in Fig. 5 we show that non-aromatic edges invariably display edge states at the Fermi level that can split up and thus induce a magnetic character of the edge. In absence of such splitting those states are only partially occupied, giving rise to metallic behavior. On the contrary, aromatic edges either do not possess edge states (as shown in Fig. 5, panel A) or such states are fully empty or occupied (as a 0 28 ).

VI. CONCLUSIONS
In conclusion, our ab initio DFT calculations indicate that neutral graphene edges, in the presence of molecular species typical of experimental conditions, are stabilized by decoration with double-bonding O/NH groups or single bonding OH/H/NH 2 groups. In contrast to the case of hydrogen saturation, for each molecular species we observe only very few possible edges in the thermodynamical stability diagram, namely an armchair edge in the case of oxygen, and one zigzag edge in the case of water and in the case of ammonia. This observation might help experimentalists in discriminating between zigzag and armchair edges and their mixtures according to the chemical composition of the atmosphere in which GNRs are obtained.
Calculations were done at IDRIS (project n • 081202). (1), (5), (6)), oxygen density (ρO = NO/(2L)) and nitrogen density (ρN = NN/(2L)) for all the studied edges. Thezo2 edge, nonmagnetic within DFT-GGA calculations, is found to be magnetic when hybrid DFT functionals are used 27 .     FIG. 5: A) Scheme and electronic band-structure of the five most stable oxygen-, water-and ammonia-terminated edges of a graphene nanoribbon. Carbon-carbon bonds are represented with the standard notation, while hydrogen, oxygen and nitrogen atoms are the small circles. The structures are periodic along the ribbon-edge with periodicity L. The gray area corresponds to the electronic-bands allowed in "bulk" graphene. The dashed line is the Fermi level. The top-right inset displays the standard representation of the aromatic carbon ring. B) Three examples of "metallic/magnetic" edges (see the definition in the text).zo2, non-magnetic within DFT-GGA calculations, is found to be magnetic when hybrid DFT functionals are used 27 ;z oh4 andão6 both magnetic. Theão6 structure is non-aromatic according to our definition, since the two small dots represent a diradical, which is the only representation compatible with the carbon chemical valence.